An amplifier comprising at least two gain stages is generally formed of an amplifier with an input transconductance followed by one or several transconductance amplifiers. An amplifier comprising at least two gain stages may have the advantage, over a single-stage amplifier, of being able to operate under low voltage while enabling an output dynamic range that can almost reach the supply voltage. A second advantage may be the possibility to obtain a high open-loop gain.
FIG. 1 schematically shows an example of an amplifier 10 with two gain stages comprising an input terminal IN and an output terminal OUT Amplifier 10 comprises a transconductance amplifier TE having a “+” input connected to input terminal IN and a “+” output terminal connected to a node F. A “+” input terminal of a transconductance inverter amplifier TS is connected to node F. The “−” output of amplifier TS is connected to output terminal OUT Call VIN, VF, and VOUT the voltages respectively at terminal IN, at node F, and at terminal OUT To ensure the loop stability, it is necessary to compensate amplifier TS. This is generally done by a so-called Miller compensation, by providing a capacitor CM between the “+” input and the “−” output of amplifier TS. Capacitor CM is generally called a Miller capacitor.
FIG. 2 shows a conventional example of an amplifier 20 with three gain stages. As compared with amplifier 10 of FIG. 1, amplifier 20 comprises an intermediary transconductance amplifier TI arranged between amplifiers TE and TS. More specifically, the “+” output of amplifier TE is connected to the “+” input of amplifier TI and the “+” output of amplifier TI is connected to the “+” input of amplifier TS. To ensure the closed loop stability of amplifier 20, an additional Miller capacitor CM′ is provided between the “+” input of amplifier TI and the “−” output of amplifier TS, in addition to the previously-described Miller capacitor CM. Such an arrangement of capacitors CM and CM′ is generally called a nested Miller structure.
The principle of the Miller compensation may be disclosed by determining in simplified fashion the transfer function of amplifier 10 shown in FIG. 1.
FIG. 3 shows an equivalent electric diagram of amplifier 10 of FIG. 1. It is desired to determine the phase variation of the transfer function of amplifier 10 at the level of the frequency of unity gain, or cut-off frequency, of amplifier 10. Such a cut-off frequency may conventionally be on the order of 1 GHz. For this purpose, a sufficient approximate of the transfer function of amplifier 10 is obtained by considering that transconductance amplifier TE is equivalent to an ideal transconductance amplifier of voltage-current gain g that charges at node F a capacitor of capacitance CL1, and that amplifier TS is equivalent to an ideal transconductance amplifier of voltage-current gain k1g that charges at terminal OUT a capacitor CL2.
In the Laplace plane, the node equation at node F can be written as follows:−gVINT+(pCL1+pCM)VF−pCMVOUT=0  (1)and the node equation at terminal OUT can be written as:(k1g−pCM)VF−(pCL2+pCM)VOUT=0  (2)
Based on relations (1) and (2), the following transfer function can be obtained:
                              -                                    V              OUT                                      V                              I                ⁢                                                                  ⁢                N                                                    =                              1                          p              ⁢                                                          ⁢                                                C                  M                                g                                              ·                                    1              -                              p                ⁢                                                                  ⁢                                                      C                    M                                                                              k                      1                                        ⁢                    g                                                                                      1              +                              p                ⁢                                                                  ⁢                                                                                                    C                                                  L                          ⁢                                                                                                          ⁢                          1                                                                    ⁢                                              C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                                      +                                                                  C                                                  L                          ⁢                                                                                                          ⁢                          1                                                                    ⁢                                              C                        M                                                              +                                                                  C                                                  L                          ⁢                                                                                                          ⁢                          2                                                                    ⁢                                              C                        M                                                                                                  g                    ⁢                                                                                  ⁢                                          k                      1                                        ⁢                                          C                      M                                                                                                                              (        3        )            
In the absence of a Miller compensation, that is, for a zero CM, relation (3) becomes:
                              -                                    V              OUT                                      V                              I                ⁢                                                                  ⁢                N                                                    =                  1                                    p              2                        ⁢                                                            C                                      L                    ⁢                                                                                  ⁢                    1                                                  ⁢                                  C                                      L                    ⁢                                                                                  ⁢                    2                                                                                                k                  1                                ⁢                                  g                  2                                                                                        (        4        )            
FIG. 4 is a Bode diagram partly showing the asymptotic behavior of gains G1 and G2 of the transfer function of amplifier 10 respectively without and with a Miller compensation and a Bode diagram showing the behavior of phase φ2 of the transfer function of amplifier 10 with a Miller compensation.
The simplified transfer function of amplifier 10 in the absence of a Miller compensation comprises a pole of second order at the origin. Pulse ω1 corresponding to the cut-off frequency of amplifier 10 with no compensation is given by the following relation:
                              ω          1                =                                                            k                1                                      ·            g                                                              C                                  L                  ⁢                                                                          ⁢                  1                                            ⁢                              C                                  L                  ⁢                                                                          ⁢                  2                                                                                        (        5        )            
The phase, not shown, of amplifier 10 with no Miller compensation, is close to −180° at the cut-off frequency (pulse ω1) so that the phase margin is close to 0°.
The simplified transfer function of amplifier 10 with a Miller compensation comprises:                first pole, called the dominant pole, at the origin;        second pole, called non-dominant pole, at pulse ω2 given by the following relation:        
                              ω          2                =                              g            ⁢                                                  ⁢                          k              1                        ⁢                          C              M                                                                          C                                  L                  ⁢                                                                          ⁢                  1                                            ⁢                              C                                  L                  ⁢                                                                          ⁢                  2                                                      +                                          C                                  L                  ⁢                                                                          ⁢                  1                                            ⁢                              C                M                                      +                                          C                                  L                  ⁢                                                                          ⁢                  2                                            ⁢                              C                M                                                                        (        6        )            capacitance CM being selected to reject the pole (pulse ω2) beyond the cut-off frequency (pulse ω4) of amplifier 10 with a Miller compensation; and                zero at pulse ω3 given by the following relation:        
                              ω          3                =                                            k              1                        ⁢            g                                C            M                                              (        7        )            
The zero being located on the right-hand half-axis of the Laplace plane, it introduces a phase drop at the same time as a gain increase. This is a conventional disadvantage of the Miller compensation and modifications of the circuit of FIG. 1 are generally implemented to reject the zero far beyond the amplifier cut-off frequency.
The dominant pole determines the cut-off frequency, corresponding to pulse ω4, of amplifier 10 with a Miller compensation enabling obtaining an appropriate phase margin MP2, for example greater than 60°. Pulse ω4 is given by the following approximate relation:
                              ω          4                ≈                  g                      C            M                                              (        8        )            
To obtain a sufficient phase margin MP2, it can be shown that the cut-off frequency (pulse ω4) of amplifier 10 with a Miller compensation is lower than the cut-off frequency (pulse ω1) of amplifier 10 with no Miller compensation.
A disadvantage of the Miller compensation thus may be that a strong decrease in the cut-off frequency (pulse ω4) of the amplifier is thus obtained. Further, the occurrence of a non-linearity due to the significant slew rate of the amplifier which results from the presence of the Miller capacitors may be observed.